Algebraic Generalized Inverse In General Linear Spaces

The theory of generalized inverses has become an important part in modern mathematics and has substantial content, such as generalized inverse of matrix, the generalized inverse of linear transformations in linear spaces, the Moore-Penrose inverse of linear operator in Hilbert space, the generalized inverse of linear operator in Banach space. The theory of perturbation and expression of generalized inverse plays a core role in generalized inverse. Generalized inverse has become an indispensable tool in solving the problem, such as the least squares problem, the ill-posed problem, the optimization problem.Let X and Y be linear spaces. Let T:D(T)(?)X â†'Y be a linear operator and T+ be its algebra generalized inverse. It is well known that generalized inverse is intimately connected with properties of algebraic complements and projectors. In 1974, M. Z. Nashed and G. F. Votruba discussed the inner inverse, outer inverse, generalized inverse in linear space. They investigated generalized inverse from the point view of pure algebra. Later, B. L. Rall and L. Kramarz considered algebra perturbation about generalized inverse.On the other hand, the perturbation of generalized inverse or Moore-Penrose generalized inverse, the bounded linear operator or closed linear operator, in Banach space or Hilbert space is essentially the additictivity of generalized inverse in linear space. Thus, the discussion on the algebra generalized inverse in the framework of linear space is more universal.In this paper, we firstly give the characteristics for I+AT+ to be bijective, and prove that if the algebra generalized inverse T+ of T= T+A has the same rang and null space as T+, then T+ must has the simplest possible expression. Next, we give a series of necessary and sufficient conditions for T+ to have the simplest possible expression. Finally, base on these result, we investigated the perturbation of generalized inverse in Banach space and Moore-Penrose generalized inverse in Hilbert space. Our results improve and extend some well known results in [7,8,21-23,25,29,41].Theorem Let T+∈L(Y,X) be an algebra generalized inverse of T∈L(X,Y),A∈L(X,Y), D(T)(?) D(A).If T=T+A has an algebra generallized inverse T+satisfying D(T+)=D(T+),R(T+)=R(T+)and N(T+)=N(T+), Then I+AT+:D(T+)â†'D(T+)be bijective and B=T+(I+AT+)-1=(I+T+A)-1T+ is an algebra generalized inverse of T.Theorem Let T+∈L(Y,X) be an algebra generalized inverse of T∈L(X,Y) A:D(A)(?) Xâ†'D(T+)be a linear operator,D(T)(?)D(A).If I+AT+:D(T+)â†'D(T+) is bijective,then the following statements are equivalent:(1)B=T+(I+AT+)-1=(I+T+A)-T+is an algebra generalized inverse of T=T+Aï¼›(2)R(T)∩N(T+)={0};(3)D(T+)=N(T+)+R(T)ï¼›(4)D(T)=R(T+)+N(T)ï¼›(5)D(T)=R(T+)+N(T)ï¼›(6)(I+AT+)-1 R(T)=R(T)ï¼›(7)(I+AT+)-1TN(T)(?) R(T)ï¼›(8)(I+T+A)-1N(T)=N(T).